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The obtained FRF were then projected at the CoM of each stage. @@ -1556,8 +1558,8 @@ exportFig('figs/ustation_comp_com_response_ry_z.pdf', 'width', 'third', 'height' #+end_subfigure #+end_figure -** Micro-station compliance - Measurement -<> +** Micro-station compliance +<> As was shown in the previous section, the dynamics of the micro-station is complex and tuning the multi-body model parameters to obtain a perfect match is hard. @@ -1622,7 +1624,7 @@ F_{\mathcal{X}} = \mathbf{J}_F^t \cdot F_{\mathcal{L}} \end{equation} Using the two Jacobian matrices, the FRF from the 10 hammer impacts to the 12 accelerometer outputs can be converted to the FRF from 6 forces/torques applied at the origin of frame $\{\mathcal{X}\}$ to the 6 linear/angular accelerations of the top platform expressed with respect to $\{\mathcal{X}\}$. -The obtained FRF from forces to linear motion are shown in Figure ref:fig:ustation_frf_compliance_xyz while the FRF from torques to angular motion are shown in Figure ref:fig:ustation_frf_compliance_Rxyz. +These FRF will be used for comparison with the Simscape model. #+begin_src matlab % Positions and orientation of accelerometers @@ -1768,68 +1770,7 @@ end FRF_cartesian = pagemtimes(Ja_inv, pagemtimes(G_raw, Jf_inv)); #+end_src -#+begin_src matlab :exports none :results none -%% Measured FRF of the compliance of the micro-station in the Cartesian frame - Translations/Forces -figure; -hold on; -plot(f, abs(squeeze(FRF_cartesian(1,1,:))), '-', 'color', colors(1,:), 'DisplayName', '$D_x/F_x$') -plot(f, abs(squeeze(FRF_cartesian(2,2,:))), '-', 'color', colors(2,:), 'DisplayName', '$D_y/F_y$') -plot(f, abs(squeeze(FRF_cartesian(3,3,:))), '-', 'color', colors(3,:), 'DisplayName', '$D_z/F_z$') -hold off; -leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1); -leg.ItemTokenSize(1) = 15; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Magnitude [m/N]'); -xlim([20, 500]); ylim([1e-9, 2e-6]); -xticks([20, 50, 100, 200, 500]) -#+end_src - -#+begin_src matlab :tangle no :exports results :results file none -exportFig('figs/ustation_frf_compliance_xyz.pdf', 'width', 'half', 'height', 'normal'); -#+end_src - -#+begin_src matlab :exports none :results none -%% Measured FRF of the compliance of the micro-station in the Cartesian frame - Rotations/Torques -figure; -hold on; -plot(f, abs(squeeze(FRF_cartesian(4,4,:))), '-', 'color', colors(1,:), 'DisplayName', '$R_x/M_x$') -plot(f, abs(squeeze(FRF_cartesian(5,5,:))), '-', 'color', colors(2,:), 'DisplayName', '$R_y/M_y$') -plot(f, abs(squeeze(FRF_cartesian(6,6,:))), '-', 'color', colors(3,:), 'DisplayName', '$R_z/M_z$') -hold off; -leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1); -leg.ItemTokenSize(1) = 15; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Magnitude [rad/Nm]'); -xlim([20, 500]); ylim([2e-7, 1e-4]); -xticks([20, 50, 100, 200, 500]) -#+end_src - -#+begin_src matlab :tangle no :exports results :results file none -exportFig('figs/ustation_frf_compliance_Rxyz.pdf', 'width', 'half', 'height', 'normal'); -#+end_src - -#+name: fig:ustation_frf_compliance -#+caption: Measured FRF of the compliance of the micro-station expressed in frame $\{\mathcal{X}\}$. Both translation terms (\subref{fig:ustation_frf_compliance_xyz}) and rotational terms (\subref{fig:ustation_frf_compliance_Rxyz}) are displayed. -#+attr_latex: :options [htbp] -#+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:ustation_frf_compliance_xyz}Compliance in translation} -#+attr_latex: :options {0.49\textwidth} -#+begin_subfigure -#+attr_latex: :width 0.95\linewidth -[[file:figs/ustation_frf_compliance_xyz.png]] -#+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:ustation_frf_compliance_Rxyz}Compliance in rotation} -#+attr_latex: :options {0.49\textwidth} -#+begin_subfigure -#+attr_latex: :width 0.95\linewidth -[[file:figs/ustation_frf_compliance_Rxyz.png]] -#+end_subfigure -#+end_figure - -** Compare with the Model -<> - -The compliance of the micro-station is extracted from the Simscape model by computing the transfer function from forces/torques applied to the positioning hexapod's top platform to the "absolute" motion of the top platform. +The compliance of the micro-station multi-body model is extracted by computing the transfer function from forces/torques applied to the positioning hexapod's top platform to the "absolute" motion of the top platform. These are compared with the measurements in Figure ref:fig:ustation_frf_compliance_model. Considering how complex the micro-station compliance dynamics is, the model compliance is matching sufficiently well the measurements for the current application. @@ -1906,7 +1847,7 @@ exportFig('figs/ustation_frf_compliance_Rxyz_model.pdf', 'width', 'half', 'heigh #+end_src #+name: fig:ustation_frf_compliance_model -#+caption: Measured FRF of the compliance of the micro-station expressed in frame $\{\mathcal{X}\}$. Both translation terms (\subref{fig:ustation_frf_compliance_xyz_model}) and rotational terms (\subref{fig:ustation_frf_compliance_Rxyz_model}) are displayed. +#+caption: Compliance of the micro-station expressed in frame $\{\mathcal{X}\}$. Measured FRF are display by solid lines, while FRF extracted from the multi-body models are shown by dashed lines. Both translation terms (\subref{fig:ustation_frf_compliance_xyz_model}) and rotational terms (\subref{fig:ustation_frf_compliance_Rxyz_model}) are displayed. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:ustation_frf_compliance_xyz_model}Compliance in translation} @@ -1923,7 +1864,7 @@ exportFig('figs/ustation_frf_compliance_Rxyz_model.pdf', 'width', 'half', 'heigh #+end_subfigure #+end_figure -* Estimation of disturbances +* Estimation of Disturbances :PROPERTIES: :HEADER-ARGS:matlab+: :tangle matlab/ustation_3_disturbances.m :END: @@ -1970,7 +1911,7 @@ Finally, the obtained disturbance sources are compared in Section ref:ssec:ustat <> #+end_src -** Measurements of disturbances +** Disturbance measurements <> **** Introduction :ignore: In this section, the ground motion disturbances is directly measured using geophones. @@ -2033,7 +1974,7 @@ ylabel('Vertical motion [$\mu$m]') #+end_src #+begin_src matlab :tangle no :exports results :results 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